How to rotate half of a plane around the split edge after cutting it in half using loop cut
Select the edge, set the 3d cursor to it (Shift+S, 4); Set the cursor as the pivot point for rotation (Dot); then select your face, and create a new custom transform orientation, it will give you the correct gizmo axes for rotation (if your edge matches any of the world axes you can skip this step); rotate as you wish. :-)
You can also use the "Active element" as your pivot point. In this case you have to select all the edges of your plane to rotate, and the "hinge" edge should be selected last (to become that active element).
And for the simple plane you can also use "Normal" as your transform orientation (it works great in this case, and you have not to deal with creating custom transform orientations).